Exact weak laws of large numbers with applications to ratios of random variables

Abstract

We study convergence in probability of weighted sums of independent random variables which are not necessarily identically distributed. The results obtained are applied to ratios of independent random variables and ratios of smallest order statistics.

A remark on the exact laws of large numbers for ratios of independent random variables

“…A direct application of (1.1) and [6, Theorem 1] yields the following exact strong law of large numbers, also appearing in [4]. THEOREM 1.1.…”

“…In the above theorem we got rid of this condition. The weak exact law under the above conditions was proved in [3]. Also ratios of order statistics attracted interest of many researchers; we refer the reader to [5,6] for further references.…”